Prerequisite Knowledge: Scalar equation for a line in 2D. Dot product and cross product of a vector.

Lines

4.1 Recognize a scalar equation for a line in two-space to be an equation of the form Ax + By + C = 0, represent a line in two-space using a vector equation (r = r0 + tm) and parametric euations, and make connections between a scalar equation, a vector equation, and parametric equations of a line in two-space

Correlating y=mx+b to r = r0 + tm*

Parametric form of a line

Vector form of a line

4.2 Recognize that a line in three-space cannotbe represented by a scalar equation, and represent a line in three-space using the scalar equations of two intersecting planes and using vector and parametric equations (e.g., given a direction vector and a point on the line, or given two points on the line)

Parametric form of line in 3D: Similarity to 2D space

Planes

4.3 recognize a normal to a plane geometrically (i.e., as a vector perpendicular to the plane) and algebraically [e.g., one normal to the plane 3x + 5y – 2z = 6 is (3, 5, –2)], and detemine, through investigation, some geometric properties of the plane (e.g., the direction of any normal to a plane is constant; all scalar multiples of a normal to a plane are also normals to that plane; three non-collinear points determine a plane; the resultant, or sum, of any two vectors in a plane also lies in the plane)

Investigation: Working with cardboard and straws, looking at equations of lines perpendicular to the plane and relation to equation of the plane

Application of lines lying in planes: Relation to dot and cross products

4.4 recognize a scalar equation for a plane in three-space to be an equation of the form Ax + By + Cz + D = 0 whose solution points make up the plane, determine the intersection of three planes represented using scalar equations by solving a system of three linear equations in three unknowns algebraically (e.g., by using elimination or substitution), and make connections between the algebraic solution and the geometric configuration of the three planes

Review intersections of lines: Relate to intersection of lines with planes and planes with planes

4.5 determine, using properties of a plane, the scalar, vector, and parametric equations of a plane

4.6 determine the equation of a plane in its scalar, vector, or parametric form, given another of these forms

Lines and Planes

4.7 solve problems relating to lines and planes in three-space that are represented in a variety of ways (e.g., scalar, vector, parametric equations) and involving distances (e.g., between a point and a plane; between two skew lines) or intersections (e.g., of two lines, of a line and a plane), and interpret the result geometrically